Quasisymmetry and unions
The paper begins with an improvement of \textit{F. W. Gehring}'s local maximum principle for quasiconformal maps [Complex Variables, Theory Appl. 5, 245-263 (1986; Zbl 0603.30024)]. In particular, the connectedness of the boundary is replaced by a much weaker condition. This result is applied to prove the following two theorems: (1) Suppose that G and \(G'\) are domains in \({\mathbb{R}}^ n\), \(n\geq 2\), and that f: \(G\to G'\) is a homeomorphism. Suppose that \(G=E_ 1\cup...\cup E_ s\) and that \(f| E_ j\) is locally \(\eta\)-quasisymmetric for each j. Then f is K-quasiconformal with \(K=\eta (1)^{n-1}\). (2) Suppose that G and \(G'\) are as above and that f: \(\bar G\to \bar G'\) is a homeomorphism such that \(f| G\) is K-quasiconformal and \(f| \partial G\) is \(\eta\)-quasisymmetric. Then f is \(\eta_ 1\)-quasisymmetric with \(\eta_ 1\) depending only on (K,\(\eta\),n). Here one can replace quasisymmetry by a quasimöbius condition.
scientific article; zbMATH DE number 4165250